In Syldavia the economists find that (annual) household consumption $c$ is related to (annual) income $y$ by the formula $c=\alpha +\beta y$, where $\alpha>0$ and $0<\beta<1$. Because of this, they argue, inequality of consumption must be less than inequality of income.
How do I provide an intuitive argument for this?
My work: 
$\frac{\partial c}{\partial y}=\beta $. So if $\beta$ increases then the $c$ curve shifts up. The change in $c$ is dependent on change in $\beta$ which is between $0$ and $1$. But I am not sure if I can intuitively explain why inequality of consumption must be less than inequality of income.